Under new 'rough radial structure' assumptions, ball averages on exponentially growing groups satisfy weak-type L(logL)^c maximal inequalities, and non-elementary hyperbolic groups satisfy the optimal weak-type (1,1) inequality.
L.,Weak-typeL 1-estimates for maximal functions on non-compact sym- metric spaces
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Hardy-Littlewood maximal operator on spaces of exponential volume growth
Under new 'rough radial structure' assumptions, ball averages on exponentially growing groups satisfy weak-type L(logL)^c maximal inequalities, and non-elementary hyperbolic groups satisfy the optimal weak-type (1,1) inequality.